Theorem seqeq1 9922

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

Step	Hyp	Ref	Expression
1		iseqeq1 9919	. 2 |- ( M = N -> seq M ( .+ , F , _V ) = seq N ( .+ , F , _V ) )
2		df-seq3 9915	. 2 |- seq M ( .+ , F ) = seq M ( .+ , F , _V )
3		df-seq3 9915	. 2 |- seq N ( .+ , F ) = seq N ( .+ , F , _V )
4	1, 2, 3	3eqtr4g 2146	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1290   _Vcvv 2620   seqcseq4 9912   seqcseq 9913

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fv 5036  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-iseq 9914  df-seq3 9915

This theorem is referenced by:  seqeq1d  9925  seq3f1olemqsum  9990  iserex  10788  isumsplit  10946  ege2le3  11022